Coded distributed computing (CDC) is a new technique proposed with the purpose of decreasing the intense data exchange required for parallelizing distributed computing systems. Under the famous MapReduce paradigm, this coded approach has been shown to decrease this communication overhead by a factor that is linearly proportional to the overall computation load during the mapping phase. Nevertheless, it is widely accepted that this overhead remains a main bottleneck in distributed computing. To address this, we take a new approach and we explore a new system model which, for the same aforementioned overall computation load of the mapping phase, manages to provide astounding reductions of the communication overhead and, perhaps counterintuitively, a substantial increase of the computational parallelization. In particular, we propose multi-access distributed computing (MADC) as a novel generalization of the original CDC model, where now mappers (nodes in charge of the map functions) and reducers (nodes in charge of the reduce functions) are distinct computing nodes that are connected through a multi-access network topology. Focusing on the MADC setting with combinatorial topology, which implies